Rock–ice avalanches are liquid–granular flows that consist of a mixture of rock, ice and a liquid. The dynamics that distinguishes these types of flows from other geophysical flows is the ice melting. This process is responsible for mass and momentum transfers between the solid and liquid components of the mixture and for the effects of lubrication and fluidization that reduce the mixture strength. In this work, we analyse the problem from a mathematical point of view. Starting from the partial differential equations of a complete three-phase approach, we identify two basic assumptions that can be used to build a framework of classes of simplified models. The implications of these assumptions on the physical description of the flow are carefully analysed for each class, and particular attention is paid to the simplification of the melting process expressed in terms of mass and momentum transfers. Moreover, the derived framework allows us to classify the existing literature models and to identify a new class of models that can be considered a reasonable trade-off between simplicity and completeness. Finally, the mathematical nature of each class is investigated by performing an in-depth analysis of the eigenvalues. Results show that the most simplified models are strictly hyperbolic, while the most complete approaches are affected by a loss of hyperbolicity in given ranges of the model parameters. Further research is necessary to understand the reasons and the numerical implications of this feature.

A rainfall threshold is a function of some rainfall quantities that provides the conditions beyond which the probability of debris-flow occurrence is considered significant. Many uncertainties may affect the thresholds calibration and, consequently, its robustness. This study aims to assess the uncertainty in the estimate of a rainfall threshold for stony debris flow based on the backward dynamical approach, an innovative method to compute the rainfall duration and averaged intensity strictly related to a measured debris flow. The uncertainty analysis is computed by performing two Monte Carlo cascade simulations: (i) to assess the variability in the event characteristics estimate due to the uncertainty in the backward dynamical approach parameters and data and (ii) to quantify the impact of this variability on the threshold calibration. The application of this procedure to a case study highlights that the variability in the event characteristics can be both low and high. Instead, the threshold coefficients have a low dispersion showing good robustness of the threshold estimate. Moreover, the results suggest that some event features are correlated with the variability of the rainfall event duration and intensity. The proposed method is suitable to analyse the uncertainty of other threshold calibration approaches.

On 27–28 July 2019, in a catchment of the Mt. Amiata area (Italy), an extreme rainfall induced a debris flow, which caused a channeled erosive process just upstream of the Abbadia San Salvatore village, the obstruction of a culvert at the entrance to the urban area, and the subsequent flooding of the village. In this paper, we present the back analysis of this event. The complexity of this case study is due to several peculiar characteristics, but above all, to the clogging of the culvert, a phenomenon difficult to simulate numerically. The methodology used for the reconstruction of the event is based on a multidisciplinary approach. A geological field investigation was carried out to characterize the catchment and assess the availability of debris. Then, a cascade of numerical models was employed to reconstruct the debris flow: the FLO-2D software was used to model the runoff along the hydrographic network while the mobile-bed debris flow TRENT2D model, available through the WEEZARD system, was used to quantify both the erosion and deposition processes that occurred during the event. To simulate the culvert clogging, a novel modelling procedure was developed and applied. Despite the challenging framework, the results, in terms of debris volume, erosion rates, deposition area, and timing of the culvert obstruction, agree reasonably well with the observed data. It is worth noticing that these results were obtained mainly using parameters set a priori, namely calibrated on a physical basis. This proves that the proposed methodology is robust and effective, with good predictive capability. Therefore, it may be considered, according to the European Union (EU) Flood Directive, an “appropriate practice and the best available technology that does not imply excessive costs” to support predictive hazard mapping of situations as the one here considered.